#9874934

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4199 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,8)(3,7,11,14)(4,5,12,13)(9,10,15,16)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34,38,41)(35,42,37,39)(36,44,46,40)(43,47,45,48);(1,3,6,11)(2,7,8,14)(4,9,12,15)(5,10,13,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,35,38,37)(34,39,41,42)(36,43,46,45)(40,47,44,48);(2,8)(4,12)(7,14)(9,15)(18,20)(19,26)(25,28)(27,32)(33,36,38,46)(34,40,41,44)(35,43,37,45)(39,47,42,48);(1,4)(2,5)(3,9)(6,12)(7,10)(8,13)(11,15)(14,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,44)(42,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,37,38,35)(34,39,41,42)(36,45,46,43)(40,47,44,48);(1,5)(2,4)(3,10)(6,13)(7,9)(8,12)(11,16)(14,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9782130);